The Morse spectrum for linear random dynamical systems

TL;DR

This paper establishes a unique Morse decomposition for linear random dynamical systems, introduces the Morse spectrum, and explores its properties, including its relation to the non-uniform dichotomy spectrum under bounded growth conditions.

Contribution

It introduces the Morse spectrum for linear random dynamical systems and proves its key properties, including its structure as a union of intervals and its relation to the dichotomy spectrum.

Findings

Morse spectrum is a finite union of closed intervals.

Unique finest weak Morse decomposition exists for these systems.

Under bounded growth, Morse spectrum equals the non-uniform dichotomy spectrum.

Abstract

We prove that projectivised finite-dimensional linear random dynamical systems possess a unique finest weak Morse decomposition. Based on this result, we define the Morse spectrum and investigate its basic properties. In particular, we show that the Morse spectrum is given by a finite union of closed intervals. Furthermore we demonstrate that under a bounded growth condition, the Morse spectrum coincides with the non-uniform dichotomy spectrum.